{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/mock-seifert-matrices-and-algebraic","title":"Mock Seifert matrices and unoriented algebraic concordance","arxiv_id":"2301.05946","date":"2023-01-14","proceeding":null,"authors":["Hans U. Boden","Homayun Karimi"],"abstract":"A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair $(K,F)$, where $K$ is a knot in a thickened surface and $F$ is an unoriented spanning surface for $K$. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted $\\mathcal{m} \\mathcal{G}^{\\mathbb Z}$ and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection $\\lambda \\colon \\mathcal{v} \\mathcal{C} \\to \\mathcal{m} \\mathcal{G}^{\\mathbb Z}$, where $\\mathcal{v} \\mathcal{C} $ denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that $\\mathcal{m} \\mathcal{G}^{\\mathbb Z}$ contains a copy of ${\\mathbb Z}^\\infty \\oplus ({\\mathbb Z}/2)^\\infty \\oplus({\\mathbb Z}/4)^\\infty.$","url_abs":"https://arxiv.org/abs/2301.05946v3","url_pdf":"https://arxiv.org/pdf/2301.05946v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"mock-seifert-matrices-and-algebraic","repo_url":"https://github.com/damianjlin/msmat","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}